The basic economics argument assumes that there are not price points that are both higher than what drivers accept and that consumers are willing to pay. The existence of Lyft plays a role here, but both are in the same boat - they either need a new product line with better margins to take off that has some synergies with the ride-sharing business, or the unit economics of ride-sharing need to get bad enough for both concurrently to decide to raise prices.
The fact that taxi cabs existed for decades and were profitable means that this price point exists. That was the whole reason lyft and uber entered the market in the first place.
It all goes in a circle though: why did city governments sanction taxi cartels (by issuing a fixed number of medallions)?
Because if anyone can be a taxi anytime (aka open market), then you get a race to the bottom, both in quality and availability. Then there isn't a steady supply and a lot of collateral problems (accidents, crime, etc). The solution a century ago was to regulate: limited number of taxis and minimum driver qualifications. In the days before apps and routing, it meant that there were taxis available most of the time, except for maybe at peak times.
My point is that taxis are an extremely good example of why you need regulation. Uber and Lyft will succeed to the point they can replace that regulation (cheap enough fares, good enough cars and drivers).
You could also argue that Uber and Lyft are controlling the taxi market because they don't let drivers set their own fare. From the point of view of the driver, taxi-driving is an Uber-Lyft-sanctioned cartel right now.
